Hyperboloid of One Sheet
The hyperboloid Of One Sheet is a surface of revolution of the curve family hyperbola.
The parametric formula for the Hyperboloid of One Sheet is:
ParametricPlot3D[{Cosh[u]*Cos[v], Cosh[u]*Sin[v], Sinh[u]}, {u, -2, 2}, {v, 0, 2*π}]
(* u → height, v → circular sweep. *)
Ruled Surface
A revolving around its transverse axis forms a surface called “hyperboloid of one sheet”. A hyperboloid is a Ruled Surface.
Ruled surfaces are surfaces that for every point on the surface, there is a line on the surface passing it. Or, in other words, a surface generated by a line. If for each point on the surface there are two lines on the surface passing it, then it's called doubly-ruled surface. Hyperboloid is a doubly-ruled surface.
Ruled surfaces also include cylinder and helicoid. There are only 3 doubly ruled surfaces: The hyperboloid, hyperbolic paraboloid, and plane.
Spinning Cube Silhouette
The silhouette of a rotating dice is a hyperbola.
Hyperboloid in Architecture
Due to its simplicity and beauty, the hyperboloid is often used in architecture for towers. They are called Hyperboloid structure.
For more photos, see: Hyperboloid Towers.
Other algebaric surfaces that has cross-sections of conic sections are: ellipsoid, paraboloid, hyperbolic paraboloid, hyperboloid of one sheet, hyperboloid of two sheets.